Boltzmann Viscoelastic Model (LAW34)

This law valid for solid elements can be used for viscoelastic materials like polymers, elastomers, glass and fluids.

Elastic bulk behavior is assumed. Air pressure may be taken into account for closed cell foams: (1)

P = - K ε_{k k} + P_{a i r}

with: (2)

P_{a i r} = - \frac{P_{0} γ}{1 + γ - Φ}; γ = \frac{V}{V_{0}} - 1 + γ_{0}

and: (3)

ε_{k k} = \ln (\frac{V}{V_{0}})

Where,

$γ$: Volumetric strain
$Φ$: Porosity
$P_{0}$: Initial air pressure
$γ_{0}$: Initial volumetric strain
$K$: Bulk modulus

For deviatoric behavior, the generalized Maxwell model is used. The shear relaxation moduli in Viscous Materials, Equation 19 is then defined as:(4)

Ψ (t) = G_{l} + G_{s} e^{- β t}

(5)

G_{s} = G_{0} - G_{l}

Where,

$G_{0}$: Short time shear modulus
$G_{l}$: Long time shear modulus
$β$: Decay constant, defined as the inverse of relaxation time $τ_{s}$ :(6)
$β = \frac{1}{τ_{s}}$ ; with $τ_{s} = \frac{η_{s}}{G_{s}}$

The coefficients

η_{s}

G_{s}

and

G_{l}

are defined for the generalized Maxwell model, as shown in Figure 1.

From Equation 4, the value of $β$ governs the transition from the initial modulus $G_{0}$ to the final modulus $G_{l}$ . For $t$ =0, you obtain $Ψ (t) \to G_{0}$ and when $t \to \infty$ , then $Ψ (t) \to G_{l}$ . For a linear response, put $G_{0} = G_{l}$ .